Precise theory of levels of hydrogen and deuterium: The Coulomb vacuum polarization correction Svetlana Kotochigova and Peter J. Mohr National Institute of Standards and Technology, Gaithersburg, MD 20899, USA (Dated: March 16, 2005) The Coulomb vacuum polarization contribution to the energy levels of hydrogen and deuterium atoms is calculated for states with n ≤ 200. The Uehling potential is the dominant contribution of the vacuum polarization to the energy levels. Numerical values for this contribution are given to nine figure precision in this paper for states with l ≤ 2. This correction is negligible for higher values of l. These results are available on the NIST Physics Laboratory Web site at physics.nist.gov/hdel. PACS numbers: 12.20.Ds, 31.30.Jv, 06.20.Jr, 31.15.-p 2 I. INTRODUCTION V = − δ ; 61 15 l0 Energy levels in hydrogen and deuterium are deter- where mined primarily by the eigenvalues of the Dirac equa- 1 if l = 0 tion. However, to obtain accurate values for the levels, δ = ; (5) l0 0 if l ≥ 1 it is necessary to take into account many additional cor- rections, including those due to quantum electrodynamic and l is the orbital angular momentum of the state. The (QED) effects. In hydrogen and deuterium, the largest (1) remaining part GVP(Zα) that arises from the Uehling QED effect is the self-energy. The next largest is the potential is readily calculated numerically, as described vacuum polarization, which is due to the creation of a in Sec. III. virtual electron-positron pair in the exchange of photons (R) The higher-order remainder GVP (Zα) has been con- between the electron and the nucleus. sidered by Wichmann and Kroll, and the leading terms in powers of Zα are [3{5] II. VACUUM POLARIZATION 19 p2 G(R)(Zα) = − δ VP 45 27 l0 The second-order vacuum-polarization level shift is 1 31p2 written as + − p(Zα)δl0 + ··· : (6) 16 2880 α (Zα)4 E(2) = H(Zα) m c2; (1) Higher-order terms omitted from Eq. (6) are negligible VP p n3 e in hydrogen and deuterium. where Z is the charge of the nucleus, which in the case of hydrogen and deuterium, is 1, α is the fine structure constant, n is the principal quantum number of the state, III. CALCULATION me is the mass of the electron, c is the speed of light, and H(Zα) is a dimensionless function of order 1. In order The numerical evaluation of the Uehling potential cor- to facilitate the calculation, that function is divided into rection has been described before [6, 7] and is reviewed in two parts, corresponding to the Uehling potential [1, 2], this section.. The Uehling potential in a Coulomb field, (1) denoted here by H (Zα) and a higher-order remainder in units where me, c, and ¯h are replaced by 1, is [1, 2] (R) (3) (5) H (Zα) = H (Zα)+H (Zα)+··· , where the super- Z 1 α 1=2 script denotes the order in powers of the external field. U(x) = − Zα dt t2 − 1 The individual terms may be expanded in a power series p 1 in Zα as 2 1 exp(−2tx) × + ; (7) (1) 2 −2 3t2 3t4 x H (Zα) = V40 + V50 (Zα) + V61 (Zα) ln(Zα) + G(1) (Zα)(Zα)2 (2) and the energy-level shift is VP Z (R) H(R)(Zα) = G (Zα)(Zα)2 ; (3) 2 VP EU = dx jφn(x)j U(x) (8) with where φ (x) is the Dirac wave function for hydrogen. 4 n V = − δ This can be written as 40 15 l0 2 Z 1 Zα 1=2 2 1 5 p E = − dt t2 − 1 + L (2t); (9) V50 = δl0 (4) U p 2 4 n 48 1 3t 3t 2 2 where ×L ; (12) Z n (2x − x2)1=2 exp(−ux) L (u) = dx jφ (x)j2 n n x Z and finally evaluating the integral with Gauss-Legendre 1 2 2 quadrature applied to the variable s, where x = s4. The = dx x f1 (x) + f2 (x) exp(−ux) ; (10) 0 values obtained here result from a 70 point evaluation, al- though this is well above the number of integration points where f1 and f2 are the radial wave functions as defined needed to obtain the quoted uncertainty. in Ref. [6] The accuracy of the two-dimensional integration was The transform Ln(u) is an elementary function for any state, and is given explicitly for the states with n = 1; 2 monitored by comparing the results of 8 × 60 point inte- gration and 10×70 point integration for each evaluation. in Ref. [6]. However, to evaluate Ln(u) for a wide range of values of n, it is convenient to numerically integrate In all cases, the difference is well below the quoted pre- the expression in Eq. (10). The integrand in Eq. (10) is cision. (1) of the form The results for GVP(Zα) are obtained by computing 1− x P (x) exp (−ax); (11) (1) −2 (1) GVP(Zα) = (Zα) H (Zα) 2 −2 where << 1 and P (x) is a polynomial with degree of −V40 − V50 (Zα) − V61 (Zα) ln(Zα) order 2n. An integral of this form is evaluated exactly by (13) applying Gauss-Laguerre quadrature of the general form that includes the fractional power in the weight function, provided a sufficient number of abscissas and weights are where H(1)(Zα) is given by employed [8]. In principle, this number is of order n in order to obtain the exact result, but in fact an accurate α (Zα)4 −1 evaluation results for a very small number of integration H(1)(Zα) = m c2 E ; (14) p 3 e U points, of order 10 or less, even when n = 200, because n the dominant contribution comes from the first few pow- IV. RESULTS ers of x. The radial wave functions f1 and f2 in Eq. (10) are readily evaluated with the aid of Ref. [9], which gives (1) The results of the numerical evaluation of GVP(Zα) for recursion relations for the polynomial terms in the wave Z = 1 and α = 1=137:036 are given in Table I. All figures functions. shown in the table are significant, and the numbers have The integral over t in Eq. (9) is evaluated by making been rounded to the nearest last figure. a change of variable to x = 1 − (1 − t−2)1=2 to obtain The Uehling potential correction for states with an- Z Zα2 1 2 + 2x − x2 gular momentum l ≥ 3 is zero to the number of figures E = − dx (1 − x)2 shown in the table. U p 2 3 0 2x − x [1] E. A. Uehling, Phys. Rev. 48, 55 (1935). [6] P. J. Mohr, Phys. Rev. A 26, 2338 (1982). [2] R. Serber, Phys. Rev. 48, 49 (1935). [7] S. Kotochigova, P. J. Mohr, and B. N. Taylor, Can. J. [3] E. H. Wichmann and N. M. Kroll, Phys. Rev. 101, 843 Phys. 80, 1373 (2002). (1956). [8] W. H. Press, et al., Numerical Recipes in FORTRAN [4] P. J. Mohr, in Beam-Foil Spectroscopy, edited by I. A. (Cambridge University Press, Cambridge, UK, 1992), 2nd Sellin and D. J. Pegg (Plenum Press, New York, 1975), ed., p. 146. vol. 1, pp. 89{96. [9] P. J. Mohr and Y.-K. Kim, Phys. Rev. A 45, 2727 (1992). [5] P. J. Mohr, At. Data. Nucl. Data Tables 29, 453 (1983). 3 (1) TABLE I: Values of the function GVP(α). n S1=2 P1=2 P3=2 D3=2 D5=2 1 −0:618 723 915 2 −0:808 872 415 −0:064 005 731 −0:014 132 445 3 −0:814 530 258 −0:075 858 689 −0:016 749 740 −0:000 000 046 −0:000 000 015 4 −0:806 578 623 −0:080 007 050 −0:017 665 786 −0:000 000 065 −0:000 000 021 5 −0:798 362 423 −0:081 927 062 −0:018 089 778 −0:000 000 075 −0:000 000 024 6 −0:791 450 317 −0:082 969 986 −0:018 320 090 −0:000 000 080 −0:000 000 026 7 −0:785 810 647 −0:083 598 809 −0:018 458 958 −0:000 000 083 −0:000 000 027 8 −0:781 196 818 −0:084 006 922 −0:018 549 086 −0:000 000 086 −0:000 000 027 9 −0:777 380 712 −0:084 286 710 −0:018 610 877 −0:000 000 087 −0:000 000 028 10 −0:774 184 470 −0:084 486 833 −0:018 655 075 −0:000 000 088 −0:000 000 028 11 −0:771 474 587 −0:084 634 896 −0:018 687 775 −0:000 000 089 −0:000 000 028 12 −0:769 151 196 −0:084 747 505 −0:018 712 646 −0:000 000 089 −0:000 000 029 13 −0:767 138 988 −0:084 835 137 −0:018 732 001 −0:000 000 090 −0:000 000 029 14 −0:765 380 486 −0:084 904 668 −0:018 747 359 −0:000 000 090 −0:000 000 029 15 −0:763 831 250 −0:084 960 760 −0:018 759 748 −0:000 000 091 −0:000 000 029 16 −0:762 456 474 −0:085 006 665 −0:018 769 887 −0:000 000 091 −0:000 000 029 17 −0:761 228 557 −0:085 044 708 −0:018 778 291 −0:000 000 091 −0:000 000 029 18 −0:760 125 367 −0:085 076 588 −0:018 785 333 −0:000 000 091 −0:000 000 029 19 −0:759 128 960 −0:085 103 566 −0:018 791 292 −0:000 000 091 −0:000 000 029 20 −0:758 224 651 −0:085 126 599 −0:018 796 380 −0:000 000 092 −0:000 000 029 21 −0:757 400 309 −0:085 146 420 −0:018 800 758 −0:000 000 092 −0:000 000 029 22 −0:756 645 832 −0:085 163 599 −0:018 804 553 −0:000 000 092 −0:000 000 029 23 −0:755 952 738 −0:085 178 586 −0:018 807 864 −0:000 000 092 −0:000 000 029 24 −0:755 313 862 −0:085 191 738 −0:018 810 770 −0:000 000 092 −0:000 000 029 25 −0:754 723 106 −0:085 203 343 −0:018 813 333 −0:000 000 092 −0:000 000 029 26 −0:754 175 248 −0:085 213 635 −0:018 815 607 −0:000 000 092 −0:000 000 029 27 −0:753 665 794 −0:085 222 804 −0:018 817 633 −0:000 000 092 −0:000 000 029 28 −0:753 190 855 −0:085 231 008 −0:018 819 445 −0:000 000 092 −0:000 000 030 29 −0:752 747 048 −0:085 238 378 −0:018 821 074 −0:000 000 092 −0:000 000 030 30 −0:752 331 416 −0:085 245 023 −0:018 822 542 −0:000 000 092 −0:000 000 030 31 −0:751 941 364 −0:085 251 035 −0:018 823 870 −0:000 000 092 −0:000 000 030 32 −0:751 574 608 −0:085 256 492 −0:018 825 076 −0:000 000 092 −0:000 000 030 33 −0:751 229 124 −0:085 261 460 −0:018 826 174 −0:000 000 092 −0:000 000 030 34 −0:750 903 117 −0:085 265 997 −0:018 827 176 −0:000 000 092 −0:000 000 030 35 −0:750 594 988 −0:085 270 149 −0:018 828 094 −0:000 000 092 −0:000 000 030 36 −0:750 303 309 −0:085 273 961 −0:018 828 936 −0:000 000 092 −0:000 000 030 37 −0:750 026 798 −0:085 277 467 −0:018 829 711 −0:000 000 092 −0:000 000 030 38 −0:749 764 303 −0:085 280 700 −0:018 830 425 −0:000 000 092 −0:000 000 030 39 −0:749 514 788 −0:085 283 688 −0:018 831 085 −0:000 000 092 −0:000 000 030 40 −0:749 277 313 −0:085 286 454 −0:018 831 697 −0:000 000 092 −0:000 000 030 41 −0:749 051 029 −0:085 289 020 −0:018 832